A022166 -id:A022166 - cp's OEIS frontend

A277218 Maximal coefficient among the polynomials in row n of the triangle of q-binomial coefficients.

1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910, 1667, 2934, 5448, 9686, 18084, 32540, 61108, 110780, 208960, 381676, 723354, 1328980, 2527074, 4669367, 8908546, 16535154, 31630390, 58965214, 113093022, 211591218, 406680465, 763535450, 1470597342, 2769176514, 5342750699, 10089240974

Offset: 0

Vladimir Reshetnikov, Oct 05 2016

nonn

q-binomial coefficients are polynomials in q with integer coefficients.

Is A055606 a shifted version of this sequence?

			Row 5 of the triangle of q-binomial coefficients is [1, 1 + q + q^2 + q^3 + q^4, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + q^2 + q^3 + q^4, 1], so the max coefficient is 2. Hence a(5) = 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
E. Friedman and M. Keith, Magic Carpets, J. Int Sequences, 3 (2000), #P.00.2.5.
Eric W. Weisstein, q-Binomial Coefficient
Wikipedia, q-binomial

Cf. A002838, A022166, A029895, A055606, A076822.

f:= proc(n) local k, c, v, q;
  uses QDifferenceEquations;
  v:= 0:
  for k from 0 to n do
    c:= coeffs(expand(expand(QBinomial(n,k,q))),q);
    v:= max(v, max(c));
  od:
v
end proc:
map(f, [$0..50]); # Robert Israel, Oct 05 2016

Table[Coefficient[Expand[FunctionExpand[QBinomial[n, Floor[n/2], q]]], q, Floor[n^2/8]], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 24 2021 *)

a(n) ~ sqrt(3) * 2^(n+2) / (Pi * n^2). - Vaclav Kotesovec, Oct 09 2016

A289539 Number of ways to choose a subspace U of GF(2)^n and then choose a subspace of U.

Original entry on oeis.org

1, 3, 12, 66, 513, 5769, 95706, 2379348, 89759799, 5188919427, 463209471288, 64236626341974, 13903296824817117, 4713694025825766861, 2510421030027019810854, 2104931848782489253483752, 2783505220978001187684672531, 5813031971452642599096778614183

Offset: 0

Geoffrey Critzer, Jul 12 2017

nonn

A q-analog (q=2) of A000244.

Cf. A000244, A182176, A289537, A289538, A289541, A289542.

nn = 20; eq[z_] := Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];
Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}] CoefficientList[ Series[eq[z]^3 /. q -> 2, {z, 0, nn}], z]

a(n) = Sum_{k=0..n} A022166(n,k)*A006116(k).

a(n)/[n]_q! is the coefficient of x^n in the expansion of exp_q(x)^3 when q -> 2 and where exp_q(x) is the q-exponential function and [n]_q! is the q-factorial of n.

A359985 Triangle read by rows: T(n,k) is the number of quasi series-parallel matroids on [n] with rank k, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 35, 15, 1, 1, 31, 155, 155, 31, 1, 1, 63, 651, 1365, 651, 63, 1, 1, 127, 2667, 10941, 10941, 2667, 127, 1, 1, 255, 10795, 82215, 156597, 82215, 10795, 255, 1, 1, 511, 43435, 589135, 1988007, 1988007, 589135, 43435, 511, 1

Offset: 0

Andrew Howroyd, Mar 08 2023

A quasi series-parallel matroid is a collection of series-parallel matroids. See the Ferroni/Larson reference for a precise definition.

The first six rows of this triangle are the same as A022166.

			Triangle begins:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,     1;
  1,  15,   35,    15,     1;
  1,  31,  155,   155,    31,    1;
  1,  63,  651,  1365,   651,   63,   1;
  1, 127, 2667, 10941, 10941, 2667, 127, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.
Nicholas Proudfoot, Yuan Xu, and Ben Young, On the enumeration of series-parallel matroids, arXiv:2406.04502 [math.CO], 2024.

Row sums are A359986.
Columns k=0..2 are A000012, A000225, A006095.
Cf. A022166, A140945.

\\ Proposition 2.3, 2.8 in Ferroni/Larson, compare A140945.
T(n) = {[Vecrev(p) | p<-Vec(serlaplace(exp(x*(y+1) + y*intformal( serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) }

A006098 Gaussian binomial coefficient [ 2n,n ] for q=2.

Original entry on oeis.org

1, 3, 35, 1395, 200787, 109221651, 230674393235, 1919209135381395, 63379954960524853651, 8339787869494479328087443, 4380990637147598617372537398675, 9196575543360038413217351554014467475, 77184136346814161837268404381760884963259795

Offset: 0

N. J. A. Sloane

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Table of n, a(n) for n = 0..35
Alin Bostan and Sergey Yurkevich, On the q-analogue of Pólya's Theorem, arXiv:2109.02406 [math.CO], 2021.
I. Siap and I. Aydogdu, Counting The Generator Matrices of Z_2 Z_8 Codes, arXiv:1303.6985 [math.CO], 2013.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A022166, A065446.

q:=2; [n le 0 select 1 else (&*[(1-q^(2*n-j))/(1-q^(j+1)): j in [0..n-1]]): n in [0..15]]; // G. C. Greubel, Mar 09 2019

Table[QBinomial[2n,n,2],{n,0,20}] (* Harvey P. Dale, Oct 22 2012 *)

q=2; {a(n) = prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1))) };
vector(10, n, n--; a(n)) \\ G. C. Greubel, Mar 09 2019

[gaussian_binomial(2*n,n,2) for n in range(0,11)] # Zerinvary Lajos, May 25 2009

a(n) = A022166(2n,n). - Alois P. Heinz, Mar 30 2016
a(n) ~ c * 2^(n^2), where c = A065446. - Vaclav Kotesovec, Sep 22 2016
a(n) = Sum_{k=0..n} 2^(k^2)*(A022166(n,k))^2. - Werner Schulte, Mar 09 2019

More terms from Harvey P. Dale, Oct 22 2012

A034674 Sum of n-th powers of divisors of 128.

Original entry on oeis.org

8, 255, 21845, 2396745, 286331153, 35468117025, 4467856773185, 567382630219905, 72340172838076673, 9241421688590303745, 1181745669222511412225, 151189550474521284184065, 19347536633519984760328193

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n=0..200
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. Cites this sequence.

Cf. A022166, A022190.

[DivisorSigma(n, 128): n in [0..20]]; // Vincenzo Librandi, Apr 17 2014

Total[#^Range[0, 20]&/@Divisors[128]] (* Vincenzo Librandi, Apr 17 2014 *)

a(n) = (2^(8*n) - 1)/(2^n - 1). Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 255*x + 43435*x^2 + ... is the o.g.f. for the 7th subdiagonal of triangle A022166, essentially A022190. - Peter Bala, Apr 07 2015

A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 2, 1, 8, 6, 1, 64, 56, 14, 1, 1024, 960, 280, 30, 1, 32768, 31744, 9920, 1240, 62, 1, 2097152, 2064384, 666624, 89280, 5208, 126, 1, 268435456, 266338304, 87392256, 12094464, 755904, 21336, 254, 1, 68719476736, 68451041280, 22638755840, 3183575040, 205605888, 6217920, 86360, 510, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

Triangle T(n,k), 0 <= k <= n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, ...] = A014236 (first zero omitted) DELTA A077957 where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006

			Triangle begins:
      1;
      2,     1;
      8,     6,    1;
     64,    56,   14,    1;
   1024,   960,  280,   30,  1;
  32768, 31744, 9920, 1240, 62, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A023531 (q=0), A007318 (q=1), this sequence (q=2), A173007 (q=3), A173008 (q=4).

Cf. A006125, A022166, A028362.

function T(n,k,q)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
  end if; return T; end function;
[T(n,k,2): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

T[n_, k_, q_]:= T[n,k,q]= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1,k,q] +T[n-1,k-1,q] ]];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

def T(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

Sum_{k=0..n} T(n, k) = A028362(n).
T(n,0) = 2^(n*(n+1)/2) = A006125(n+1). - Philippe Deléham, Nov 05 2006
T(n,k) = 2^binomial(n+1-k,2) * A022166(n,k) for 0 <= k <= n. - Werner Schulte, Mar 25 2019

A270882 Triangle read by rows: D*(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2) with a block containing any given nonzero vector.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 16, 12, 0, 1, 176, 560, 224, 0, 1, 3456, 40000, 53760, 13440, 0, 1, 128000, 5848832, 20951040, 15554560, 2666496, 0, 1, 9115648, 1934195712, 17826414592, 30398054400, 14335082496, 1791885312, 0, 1, 1259921408, 1510821195776, 37083513880576, 134908940386304, 133854174117888, 43693331447808, 4161269661696

Offset: 0

Michel Marcus, Mar 25 2016

			Triangle begins:
  1;
  0, 1;
  0, 1,      2;
  0, 1,     16,      12;
  0, 1,    176,     560,      224;
  0, 1,   3456,   40000,    53760,    13440;
  0, 1, 128000, 5848832, 20951040, 15554560, 2666496;
  ...

Jinyuan Wang, Rows n = 0..10 of triangle, flattened
David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
David Ellerman, The Quantum Logic of Direct-Sum Decompositions, arXiv preprint arXiv:1604.01087 [quant-ph], 2016. See Section 7.5.

Cf. A270880, A270883 (row sums).

The main diagonal appears to match A377642. - Nikita Babich, Nov 17 2024

(* about 40 seconds on a laptop computer *) g[n_] := q^Binomial[n, 2] * FunctionExpand[QFactorial[n, q]]*(q - 1)^n /. q -> 2; d[k_, m_] :=Map[PadRight[#, 10] &,Table[Table[Total[Map[g[n]/Apply[Times, g[#]]/Apply[Times, Table[Count[#, i], {i, 1, n}]!] &,IntegerPartitions[n, {j}]]], {j, 1, n}], {n, 1, 10}]][[k, m]];d[0, m_] := If[m == 0, 1, 0]; d[k_, 0] := If[k == 0, 1, 0];s[n_, m_] :=Sum[FunctionExpand[QBinomial[n - 1, k, 2]]*2^(k (n - k))*d[k, m - 1], {k, 0, n - 1}]; Table[Table[s[n, m], {m, 1, n}], {n, 1,7}] (* Geoffrey Critzer, May 20 2017 *)

Recurrence: a(n) = Sum_{k=0..n-1} q-binomial(n-1,k)*q^(n*(n-k))*D_q(k,m-1) where D_q(k,m-1) is given by A270880 for q = 2 and where the q-binomial for q = 2 is given by A022166. This formula is the q-analog of summation formula for the Stirling numbers of the second kind A008277 so when q = 1, it reduces to that formula. - David P. Ellerman, Mar 26 2016

Name extended by David P. Ellerman, Mar 26 2016

Row 8 from Geoffrey Critzer, May 20 2017

A289537 Triangle read by rows: T(n,k) is the number of k-dimensional subspaces of an n-dimensional vector space over F_2 that do not contain a given nonzero vector, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 4, 0, 1, 14, 28, 8, 0, 1, 30, 140, 120, 16, 0, 1, 62, 620, 1240, 496, 32, 0, 1, 126, 2604, 11160, 10416, 2016, 64, 0, 1, 254, 10668, 94488, 188976, 85344, 8128, 128, 0, 1, 510, 43180, 777240, 3212592, 3108960, 690880, 32640, 256, 0

Offset: 0

Geoffrey Critzer, Jul 07 2017

			Triangle begins:
  1;
  1,    0;
  1,    2,    0;
  1,    6,    4,    0;
  1,   14,   28,    8,    0;
  1,   30,  140,  120,   16,    0;
  1,   62,  620, 1240,  496,   32,    0;

Cf. A022166, A182176, A289538, A289539, A289541, A289542.

Table[Table[Product[q^n - q^i, {i, 1, k}]/Product[q^k - q^i, {i, 0, k - 1}] /. q -> 2, {k, 0, n}], {n, 0, 9}] // Grid

T(n,k) = 2^k * A022166(n-1,k).

A157638 Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 2.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 60, 210, 60, 1, 1, 155, 1550, 1550, 155, 1, 1, 378, 9765, 27900, 9765, 378, 1, 1, 889, 56007, 413385, 413385, 56007, 889, 1, 1, 2040, 302260, 5440680, 14055090, 5440680, 302260, 2040, 1, 1, 4599, 1563660, 66194940

Offset: 0

Roger L. Bagula, Mar 03 2009

Other triangles in the family (see name) include: q = 2 (this triangle), q = 3 (see A157640), and q = 4 (see A157641). - Werner Schulte, Nov 16 2018

			Triangle begins:
  1;
  1, 1;
  1, 6, 1;
  1, 21, 21, 1;
  1, 60, 210, 60, 1;
  1, 155, 1550, 1550, 155, 1;
  1, 378, 9765, 27900, 9765, 378, 1;
  1, 889, 56007, 413385, 413385, 56007, 889, 1;
  1, 2040, 302260, 5440680, 14055090, 5440680, 302260, 2040, 1;
  1, 4599, 1563660, 66194940, 417028122, 417028122, 66194940, 1563660, 4599, 1;

Andrew Howroyd, Rows n=0..49 of triangle, flattened

Cf. A007318, A005329, A022166, A157640, A157641.

q:=2; [[k le 0 select 1 else Binomial(n,k)*(&*[(1-q^(n-j))/(1-q^(j+1)): j in [0..(k-1)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 17 2018

t[n_, m_] = Product[Sum[k*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}];
b[n_, k_, m_] = t[n, m]/(t[k, m]*t[n - k, m]);
Flatten[Table[Table[b[n, k, 1], {k, 0, n}], {n, 0, 10}]]

T(n,k) = {binomial(n,k)*polcoef(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)} \\ Andrew Howroyd, Nov 14 2018

q=2; for(n=0,10, for(k=0,n, print1(binomial(n,k)*prod(j=0,k-1, (1-q^(n-j))/(1-q^(j+1))), ", "))) \\ G. C. Greubel, Nov 17 2018

[[ binomial(n,k)*gaussian_binomial(n,k).subs(q=2) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Nov 17 2018

T(n,k) = t(n)/(t(k)*t(n-k)) where t(n) = Product_{k=1..n} Sum_{i=0..k-1} k*2^i.
T(n,k) = binomial(n,k)*A022166(n,k) for 0 <= k <= n. - Werner Schulte, Nov 14 2018
T(n,k) = n!*A005329(n)/(k!*A005329(k)*(n-k)!*A005329(n-k)). - Andrew Howroyd, Nov 14 2018

Edited and simpler name by Werner Schulte and Andrew Howroyd, Nov 14 2018

A157784 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (4^(i-1)-x), in row n and column 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 4, -5, 1, 64, -84, 21, -1, 4096, -5440, 1428, -85, 1, 1048576, -1396736, 371008, -23188, 341, -1, 1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1, 4398046511104, -5863704100864, 1563272675328, -99158478848

Offset: 0

Roger L. Bagula, Mar 06 2009

Row sums except n=0 are zero.
The matrix inverses seem to be related to the Gaussian q-form combinations.
Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=4)=[1,3,16,60,256,1008,4096,16320,65536,261888,...] DELTA [ -1,0,-4,0,-16,0,-64,0,-256,0,-1024,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2009

			Triangle begins
  1;
  1, -1;
  4, -5, 1;
  64, -84, 21, -1;
  4096, -5440, 1428, -85, 1;
  1048576, -1396736, 371008, -23188, 341, -1;
  1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1;
  4398046511104, -5863704100864, 1563272675328, -99158478848, 1549351232, -5963412, 5461, -1;
  72057594037927936, -96075326035066880, 25618523216674816, -1626175790120960, 25483729063936, -99253893440, 95436436, -21845, 1;
Row n=3 represents 64 - 84*x + 21*x^2 - x^3.

Cf. A135950, A022166, A157783, A157832.

A157784 := proc(n,k)
    product( 4^(i-1)-x,i=1..n) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

Clear[f, q, M, n, m];
q = 4;
f[k_, m_] := If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[k == n && m > 1, 1, 0]]]];
M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
Table[M[n], {n, 1, 10}];
Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
Flatten[a]

cp's OEIS Frontend

A277218 Maximal coefficient among the polynomials in row n of the triangle of q-binomial coefficients.

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A289539 Number of ways to choose a subspace U of GF(2)^n and then choose a subspace of U.

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A359985 Triangle read by rows: T(n,k) is the number of quasi series-parallel matroids on [n] with rank k, 0 <= k <= n.

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A006098 Gaussian binomial coefficient [ 2n,n ] for q=2.

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A034674 Sum of n-th powers of divisors of 128.

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A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

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A270882 Triangle read by rows: D*(n,m) is the number of direct-sum decompositions of a finite vector space of dimension n with m blocks over GF(2) with a block containing any given nonzero vector.

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